Recall the law \(\mathbb {P}^{{q}}_{n,K_{n}}\)  on maps with \(n-1\)  edges and \(K_{n}+1\)  vertices, induced by a sequence of weights \({q}= (q_i ; i \geqslant 1)\) . Property (ii) above allows us to determine the law of the associated random labelled tree.
In the context of the bijection from [1]} only, this was first discussed by Marckert & Miermont [2]}, and it was then adapted using bijection from [3]} in [4]}, see the proof of Proposition 11 in the latter reference.
Recall the set of bridges \({B}_{k}\)  as defined in (REF ); its cardinality is the binomial factor \(\binom{2k-1}{k-1}\) . Then define from \({q}\)  a weight sequence \({\theta } = (\theta (i) ; i \geqslant 0)\)  by
\(\theta (0)=1 \qquad \text{and}\qquad \theta (i) = \binom{2i-1}{i-1} q_{i} \qquad \text{for every }i \geqslant 1.\) 
